
where the optimization variable \(X\)  is constrained to be positive semidefinite (psd), i.e., \(X \in \mathbb {S}^n_+\) . That is, linear optimization over the cone of PSD matrices can be addressed by a semidefinite program. Although many algorithms, such as interior point methods, could solve SDPs to arbitrary accuracy in polynomial time, scaling as \(O(n^2m^2+n^3m) \)  per iteration [1]}. In practice, when the PSD cone is high dimensional, the solving time and memory requirements tend to be impractical. More specifically, when \(m\)  is fixed, the computational cost of solving an SDP is primarily determined by the size of the largest semidefinite constraint \(n\) .
This motivates the study of methods to decompose, or approximate high-dimensional PSD constraints into conic
combinations of smaller PSD cones. Such alternatives allow one to trade off computation time with solution quality.
In this section, we first review several inner approximation techniques of the PSD cone \(\mathbb {S}^n_+ \) . Then we provide a novel PSD cone decomposition framework by means of conical combinations of smaller dimensional subcones to unify these results.
